Seminar za biomatematiko in matematično kemijo - Arhiv

Nanotubical graphs are obtained by wrapping a hexagonal grid into a cylinder, and then possibly closing the tube with patches. Here we consider the asymptotic values of Wiener, generalized Wiener, Schultz (also known as degree distance), Gutman, Balaban, Sum-Balaban, and Harary indices for (all) nanotubical graphs of type (k, l) on n vertices. First, we determine the number of vertices at distance d from a particular vertex in an open (k, l) nanotubical graph. Surprisingly, this number does not depend much on the type of the nanotubical structure, but mainely on its circumference. At the same time, the size of a cap of a closed (k, l)-nanotube is bounded by a function that depends only on k and l, and that those extra vertices of the caps do not influence the obtained asymptotical value of the distance based indices considered here. Consequently the asymptotic values are the same for open and closed nanostructures. Finally, we obtained that the leading term of all considered topological indices depends on the circumference of the nanotubical graph, but not on its specific type. Thus, we conclude that these distance based topological indices seem not to be the most suitable for distinguishing nanotubes with the same circumference and of different type as far as the leading term is concerned.

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The canonical double cover (CDC) of a graph G is the direct product G × K₂. Graphs with the same CDC share the same walk matrix but not necessarily the same main eigenvalues or eigenvectors that determine the number of walks between pairs of vertices. We explore a new concept to see to what extent the main eigenspace determines the entries of the walk matrix of a graph. We establish a hierarchy of inclusions connecting classes of graphs in view of their CDC, walk matrix, main eigenvalues and main eigenspaces. We provide a new proof that graphs with the same CDC have two–fold symmetry and are characterized as TF-isomorphic graphs. In the source and sink potential (SSP) model, current flowing through the bonds of a Pi system molecule, from the source atom to the sink one, may choose a shortest path or may take a longer route, possibly flowing along the edges of cycles. Molecular electronic devices with the same CDC are likely to offer the same resistance to current flow for corresponding terminals.

Keywords: bipartite (canonical) double covering, main eigenspace, comain graphs, walk matrix, two-fold isomorphism, SSP model.

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